| L(s) = 1 | + (1.20 − 3.70i)2-s + (−5.80 − 4.22i)4-s + (−3.34 + 10.6i)5-s + 6.27·7-s + (2.58 − 1.87i)8-s + (35.5 + 25.2i)10-s + (13.6 − 41.9i)11-s + (−13.2 − 40.7i)13-s + (7.56 − 23.2i)14-s + (−21.5 − 66.4i)16-s + (81.6 − 59.3i)17-s + (−65.1 + 47.3i)19-s + (64.4 − 47.8i)20-s + (−139. − 101. i)22-s + (48.5 − 149. i)23-s + ⋯ |
| L(s) = 1 | + (0.425 − 1.31i)2-s + (−0.726 − 0.527i)4-s + (−0.298 + 0.954i)5-s + 0.339·7-s + (0.114 − 0.0829i)8-s + (1.12 + 0.797i)10-s + (0.373 − 1.15i)11-s + (−0.282 − 0.870i)13-s + (0.144 − 0.444i)14-s + (−0.337 − 1.03i)16-s + (1.16 − 0.846i)17-s + (−0.786 + 0.571i)19-s + (0.720 − 0.535i)20-s + (−1.34 − 0.979i)22-s + (0.440 − 1.35i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.948544 - 1.96110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.948544 - 1.96110i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (3.34 - 10.6i)T \) |
| good | 2 | \( 1 + (-1.20 + 3.70i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 - 6.27T + 343T^{2} \) |
| 11 | \( 1 + (-13.6 + 41.9i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (13.2 + 40.7i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-81.6 + 59.3i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (65.1 - 47.3i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-48.5 + 149. i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-199. - 145. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-69.6 + 50.6i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-61.0 - 188. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (17.2 + 53.1i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (85.9 + 62.4i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-153. - 111. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-140. - 432. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-171. + 528. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (773. - 561. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (454. + 330. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (54.7 - 168. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (769. + 558. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (11.3 - 8.23i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-143. + 440. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.05e3 - 764. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.52624487743885926522348550884, −10.54511597501999833087773267888, −10.16982633154202126724079091209, −8.574682494867938828800998221923, −7.47101887102277400708806514869, −6.18399134047686036663523569054, −4.71919950510029094974219301104, −3.37958191661285017332028540018, −2.69665551328559042881468661703, −0.858080805123662528769977544782,
1.64008005518493358562630872603, 4.17931864941140793077656949427, 4.83729311908622822391114177954, 5.97294084964647814491549035863, 7.12552213418623125507343279485, 7.924941376113002258665742205485, 8.875451200023508241307201195704, 9.950854002159003589833419637244, 11.43893636956918330986200555939, 12.34811741582627722701168468558
